Mohr’s circle & stress transformation worksheet with answers

Free printable Mohr’s circle & stress transformation worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy
σx​ = −10σy​ = −70MPaprincipal element

At a point in a loaded part the in-plane principal stresses are σ₁ = −10 MPa and σ₂ = −70 MPa (tension positive). Find the normal stress σavg that acts on the planes of maximum in-plane shear stress.

σ1 = −10 MPaσ2 = −70 MPa
σavg = MPa
2.Easy
σx​ = 130σy​ = −70MPaprincipal element

At a point in a loaded part the in-plane principal stresses are σ₁ = 130 MPa and σ₂ = −70 MPa (tension positive). Find the maximum in-plane shear stress τmax.

σ1 = 130 MPaσ2 = −70 MPa
τmax = MPa
3.Medium
σx​ = 120σy​ = −80τxy​ = −50MPa

A point has σₓ = 120 MPa, σᵧ = −80 MPa and τₓᵧ = −50 MPa (tension positive). The element is rotated 25° counter-clockwise. Find the normal stress σy′ on the faces of the rotated element that are perpendicular to the rotated y′-axis.

σx = 120 MPaσy = −80 MPaτxy = −50 MPaθ = 25°
σy′ = MPa
4.Medium
σx​ = −10σy​ = 70τxy​ = −20MPa

A point in a loaded part has σₓ = −10 MPa, σᵧ = 70 MPa and τₓᵧ = −20 MPa (tension positive). Find the angle θp from the x-axis to the direction of the largest principal stress, between −90° and 90°.

σx = −10 MPaσy = 70 MPaτxy = −20 MPa
θp = °
5.Hard
σx​ = 60σy​ = 50τxy​ = −30MPa

A point in a loaded part has σₓ = 60 MPa, σᵧ = 50 MPa and τₓᵧ = −30 MPa (tension positive). Find the shear stress τθ on a plane whose normal is rotated 25° counter-clockwise from the x-axis (positive in the same sense as τₓᵧ).

σx = 60 MPaσy = 50 MPaτxy = −30 MPaθ = 25°
τθ = MPa
6.Hard
σx​ = 100σy​ = 10τxy​ = 30MPa

A point in a loaded part has σₓ = 100 MPa, σᵧ = 10 MPa and τₓᵧ = 30 MPa (tension positive). Find the shear stress τθ on a plane whose normal is rotated 15° counter-clockwise from the x-axis (positive in the same sense as τₓᵧ).

σx = 100 MPaσy = 10 MPaτxy = 30 MPaθ = 15°
τθ = MPa

Answer key

  1. 1. σavg = −40 MPa
  2. 2. τmax = 100 MPa
  3. 3. σy′ = −5.977 MPa
  4. 4. θp = −76.72 °
  5. 5. τθ = −23.11 MPa
  6. 6. τθ = 3.481 MPa

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